The Möbius Strip; A Lab Activity 2

Teacher's preparation:

This is designed to follow the first
moebius strip lab. If you
are not using that lab, read the introductory material for it before
using this lab.

Materials needed:
For convenience I use adding machine tape cut in 60 cm lengths.
But any paper (including newspaper) which can be cut and pasted into
strips will work.

Tape or glue stick for each group.

Scissors for each group.Student Introduction:

How many of you were surprised when we cut the
Moebius strip
and it was still one loop?

Today we are going to find out about some other
properties
of the Moebius strip, and other loops as well. First, we will
learn some technical terms which will help us to
describe
what we observe. For those who like to
impress their parents, these terms are from the science
of topology.
Topology is the study of surfaces.

We'll use this ordinary piece of paper to explain. This paper
has two surfaces, we usually call
them the front side and the back side.
But the word "side" has so many uses it could be
confusing. This paper has one edge, we
usually refer to them as the sides and top and bottom of the
page. In topology an edge is the sharp break
between surfaces. It is not just a
corner, if we fold the surface around a corner, it is still only one
surface. The edge is the limit of a
surface.

If you can run your finger along an edge, better
use a
pencil and avoid paper cuts, and come back to where you started without
crossing a surface, as I have just done, then it is one edge,
regardless of
corners.So, this piece of paper has
only one edge.

Next, if you can draw a line from point A to point
B without
crossing an edge, A and B are on the same surface.If the line from A to C has to cross an edge, then A and C are
on different surfaces.

Does that make sense? Any qusetions yet?

Today each group will study three loops and
compare the
observations.

At the signal: Gofers get the materials, 3 strips
of paper, scissors, glue stick

Directors pick up the instructions

Now

Directions loop A

1. With the first
strip of paper, make a loop with out
any twist at all.

Describe the loop on your paper, how many half
twists does it have?

2. Place the loop on
a corner of the table so that part of it lies flat on the table with
the loop hanging below.

3. One will hold a
pencil against the paper (near the center) to make a line.

4. Another will pull
the loop smoothly under the pencil until the line joins itself.

Since the line did not cross any edges, it is all
on one surface of the loop. Is there any
surface with no line on it?

Add to your description, how many surfaces does
it
have? How do you know?

5. In a similar way
find out how many edges this loop has.

Add to your description, how many edges does it
have? How do you know?

6. Cut the loop along the center line.

Describe the result, surfaces, edges, twists,
anything else that might be of interest.

Directions loop B

1. With the second
strip of paper, make a loop with one
half twist.

Describe the loop on your paper, how many half
twists does
it have?

2. Place the loop on
a corner of the table so that part of it lies flat on the table with
the loop hanging below.

3. One will hold a
pencil against the paper (near the center) to make a line.

4. Another will pull
the loop smoothly under the pencil until the line joins itself.

Since the line did not cross any edges, it is all
on one
surface of the loop. Is there any surface with no line on it?

Add to your description, how many surfaces does it
have? How do you know?

5. In a similar way
find out how many edges this loop has.

Add to your description, how many edges does it
have? How do you know?

6. Cut the loop along the center line.

Describe the result, surfaces, edges, twists,
anything else that might be of interest.

7. Predict: What do
you think the loop would be like if you cut down the center again?

8. Now find out.

Describe the result, surfaces, edges, twists,
anything else that might be of interest.

Directions loop C

1. With the third
strip of paper, make a loop with two half twists.

Describe the loop on your paper, how many half
twists does it have?

2. Place the loop on
a corner of the table so that part of it lies flat on the table with
the loop hanging below.

3. One will hold a
pencil against the paper (near the center) to make a line.

4. Another will pull
the loop smoothly under the pencil until the line joins itself.

Since the line did not cross any edges, it is all
on one
surface of the loop. Is there any
surface with no line on it?

Add to your description, how many surfaces does it
have? How do you know?

5. In a similar way
find out how many edges this loop has.

6. cut loop along the center line.

Describe the result, surfaces, edges, twists, anything else that
might be of interest.

7. Predict: What do
you think the loop would be like if you cut down the center again?

8. Now find out.

Describe the result, surfaces, edges, twists,
anything else
that might be of interest.